2 Reference Systems Reference systems are introduced in order to model geodetic observations as a function of unknown parameters of interest. The coordinate systems are defined in terms of orientation, metrics, and curvature; they are three-dimensional in principIe (HEITZ1988). A fourth dimension, time, enters through the mutual motion of the earth and other celestial bodies and through the earth's deformations. As with the earth, reference systems can be defined for the moon and the planets in the solar system. Basic units and fundamental constants are foundational to the geodetic measurement and modeling process [2.1]. Time systems are based either on processes of quantum physics or on the daily rotation of the earth [2.2 ]. Global reference systems are realized through reference frames, e.g., the ITRF, which is established and maintained by the Intemational Earth Rotation Service [2.3]. We distinguish between the space-fixed celestial reference system [2.4] and the earth-fixed terrestrial reference system [2.5] (KovALEVSKYet al. 1989). In addition, gravity related reference systems have to be introduced, as most geodetic observations refer to the earth's gravity field [2.6]. 2.1 Basic Units and Fundamental Constants Length, mass, and time are basic quantities used in geodesy. The units for these quantities are the meter (m), the kilogram (kg), and the second (s) respectively. They are defined through the Intemational System of Units (Systeme Intemational d'Unités SI), established in 1960 by the 11th General Conference of Weights and Measures (CGPM) in Paris (MARKOWITZ1973, BIPM 1991). The definitions are as follows: . The meter is the length of the path traveled by light in vacuum during a time interval of 1/299792458 of a second (CGPM 1983). The kilogram is the unit of mass; it is equal to the mass of the intemational prototype of the kilogram (CGPM 1901). The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133atom (CGPM 1967). The establishment and maintenanceof the reference standards for these units is the task of the Bureau lnternational des Poids et Mésures (BIPM), located in Sevres, France. BIPM cooperates with the national laboratories of standards under the guidelines of the Intemational Meter Convention (1875). These national laboratories include the National Institute af Standards and 2.1 Basie Dnits and Fundamental Constants 19 Teehnology, Gaithersburg, Md., U.S.A.; the National Physical Laboratory, Teddington, u.K.; and the Physikaliseh-Teehnisehe Bundesanstalt, Braunsehweig, Germany. The realization of the meter is based on interferometrie measurements (relative uneertainty 10-12)using light with highly stable frequencies (stabilized lasers). The intemational kilogram prototype has been kept in BIPM sinee 1889; national prototypes are related to it with an uneertainty of 10-9. The BIPM Time Seetion (untilI987: Bureau Intemational de l'Heure Blli, Paris) defines the second (relative uneertainty 10-14)and the atomie time seale, ef. [2.2.1]. Previous definitions of the meter and the seeond were based on natural measures. The meter was intended to be one ten-rnillionth part of the meridian quadrant passing through Paris. Its length was derived from an are measurement,ef. [1.3.2], and realized in 1799 by a prototype meter bar ealled "metre des arehives" (legal meter). Following the International Meter Convention, a more stable version (platinium-iridium bar) was manufaetured (international meter). It has been preserved sinee 1889 at the BIPM. This definition (uneertainty 10-7) was valid until1960 when, for the first time, the wavelength of a eertain speetralline of light beeame the defining quantity. Sinee ancient times, the natural measure for time has been the daily rotation of the earth about its axis. The mean solar day, cf. [2.2.2], was determined by astronornie observations, and the seeond was defined as 1/86400 part of that day. From the 1930's on, it beeame obvious that this definition was uneertain by about 10-7 due to the irregularitiesof the earth's rotation, cf. [2.5.2]. As a supplementary SI unit, the radian (rad) is used for plane angles : . The radian is the plane angle between two radii of a circ1e subtended by an are on the eireumferenee having a length equal to the radius. Geodesy, astronomy, and geography aIso use the sexagesimal graduation with 1 full circ1e = 360° (degrees), 1° = 60' (minutes), and l' = 60" (seeonds, also aresee). With 21l:rad eorresponding to 360°, an angle ais transformed from radian to degree by a(o)= p(o)arad, pO =180°/71:. (2.1) Among the fundamental constants used in geodetic models is the velocity of light in a vaeuum, whieh is by definition (1983) c =299792458ms'l , (2.2) and the gravitational constant (CODATA system of physieal eonstants 1986), whieh is defined as 20 2 Reference Systems G =(6.672 59 :t0.000 85)xlO-Um3 kg-1S-2. (2.3) Cavendish carried out the first experimental determination of G in 1798 with a torsion balance. Current work concentrates on increasing the relative accuracy of G to better than 10-4.This incIudes investigations into dependence of G on material, external influences, distance and direction, as welI as non-inverse-square properties of gravitation (GIlliES 1987, FISCHBACHand T ALMADGE 1999). Other units and constants used in geodesy, astronomy, and geophysics will be introduced in the corresponding chapters, see also AHRENS(1995), BURSA (1995), GROTEN(2000). 2.2 Time Systems Time plays a fundamental role in geodesy. This is due to the fact that most measurement methods use the signal traveI-time of electromagnetic waves for positioning, and that a uniform time scale is also needed in order to model the motion of artificial satellites. On the other hand, a time system is required for describing the relative motion of the earth in the solar system with respect to inertial space and for describing earth deformations due to internal and external forces. Time systems are defined by the unit for a time interval and by a time epoch. They are based either on the definition of the SI second [2.2.1] or on the diurnal rotation of the earth about its axis [2.2.2]. Fundamental descriptions of time systems are found in MUELLER(1969), MORITZand MUELLER(1987), SEIDELMANN(1992). 2.2.1 Atomic Time, Dynamical Time A uniform time-scale of high accuracy is provided by the lnternational Atomic Time (Temps Atomique International: TAl). It corresponds to the definition of the SI second, ct: [2.1], which has been made approximately equal to the second of the former1yused ephemeris-time. The latter was defined by the motion of the earth about the sun and determinedthrough long-term astronomic observations. The origin of TAl was chosen so that its epoch (January 1, 1958, O h) coincided with the corresponding epoch of Universal Time UTl, cf. [2.2.2]. The TAl day comprises 86400s, and the Julian Century has 36525 TAl days. TAl is realized by a large set (more than 200) of atomic c10cks (most1y cesium beam frequency standards providing long-term stability and a few hydrogen 2.2 Time Systems 21 masers) maintained at about 60 laboratories around the world. Clock comparlsons are performed at a number of timing centers, employing GPS observations for time links, cf. [5.2.5]. From these local determinations, a weighted mean is calculated at the BIPM Time Section. The relative frequency- stability of TAl is between a few 10-1\over minutes to days) and 10-13(over years). Due to relativistic effects, the readings of the atomic c10cksare reduced to a commonheight reference (SI second "on the geoid"). The motions of ce1estialbodies and artificial satellites have to be described by a strict1yuniform time scale (inertial time). This is provided by a dynamical time, which is based on motions of bodies in the solar system. Dynamical time scales refer either to the barycenter of the solar system (Barycentric Dynamic Time TDB) or to the geocenter (Terrestrial Time TT). The TT unit is practically equivalent to TAl, with a constant difference resulting from the epoch definition ofTAI: TT =TAl + 32.184s. (2.4) Dynamical time is used in celestial mechanics with Newton's equations of motion, e.g., as an argument for the astronomical ephemeris of the moon and the sun. 2.2.2 Sidereal aDd UDiversal Time The diurna! rotation of the earth provides a natural measure for time. Corresponding time systems are introduced in order to relate earth-based observations to a space-fixed system: Sidereal and Universal (solar) Time. Hereby, two periodic motions of the earth play a role (Fig. 2.1): NORTH POLE EQUATORIAL PLANE Fig. 2.1 Earth rotation, equatorial plane, and ec1iptic plane 22 2 ReferenceSystems . the diurnal rotation (spin) of the earth about its polar axis. This rotational axis approximately coincides with the axis of maximum moment of inertia, and it passes through the earth's center of mass, cf. [2.5.2]. The equatorial plane is perpendicular to the axis of rotation, . the annual revolution of the earth around the sun. Following Kepler's laws, the earth describes an ellipse with the sun at one of its focal points. Minor perturbations arise due to the gravitation of the moon and other planets. The plane of the earth's orbit is called the ecliptic plane; it has an obliquity of about 23.5° with respect to the equatorial plane. By circumscribing the unit sphere around the center of the earth, simple geometric relations are obtained. The celestial equator and the ecliptic are defined by the intersections of the sphere with the corresponding planes.